It depends. Vehicle 'explosion' damage is calculated... strangely, in Just Cause 2. More likely than not, your car would be fine at the bottom. When you're IN a vehicle, it becomes much more resistant to damage (with the exception of planes, which become less resistant). Have you noticed that jumping out of a moving vehicle tends to cause it to explode when it hits anything, even if it's going very slowly?
In any case, I have had vehicles catch fire and even explode on very rare occasion after long drops with me still inside, but in most of those cases, the vehicle was already damaged. I've driven off of that particular cliff hundreds of times, and almost always make it to the bottom intact.
If I had to pick something to give as an answer, I'd say that you're much more likely to explode if you land directly (with all four wheels) on a flat surface from a pretty substantial height. Let me explain why I think this is:
Vehicle damage and vehicle instant explosion appear to behave as two completely different things, to me. Vehicles appear to actually fall pretty slowly, and not accelerate, but I think there's a game 'value' or something for impact that keeps increasing as you fall, to a point. However, I get the impression that this 'value' only affects the vehicles instant explosion, and the actual falling speed is what affects vehicle damage. So, if you fall from a great height, but not quite great enough for the instant explosion to be triggered, your vehicle will only receive a (relatively slight) amount of damage.
Have you tried driving a vehicle off a cliff so that it rolls down, bouncing against the cliff face on the way down? This almost always destroys your vehicle before reaching the bottom (on a good sized cliff) by causing too much damage to it, NOT by causing it to explode instantly. Since the vehicle is in more contact with the hillside on the way down than it would be if you didn't touch until you hit the bottom, the vehicle suffers more damage, even though, logically, it should be falling so fast as to be utterly obliterated in the latter scenario.
(I'm not sure how accurate this stuff about falling 'values' is, but it's the best way I can think of to explain it right now. This is drawing from my hundreds of hours of game play time. I can tell you right now that however this system works, it's extremely inconsistent. Vehicles seem to arbitrarily explode in any situation.)
The reason this 'doesn't apply to motorbikes', as you say, is because the only time you ever land a motorbike with you still on it is on a flat surface: otherwise you always fall off, and the bike then becomes susceptible to imminent explosion. This also means it does apply to boats.
Try this: hit the gas in a fast boat and aim directly for a small island. You'll clear the entire island with little damage. But if you jump out of the boat first, it will detonate as soon as it hits the beach.
Not sure how helpful this is, but I found it interesting to drive some vehicles over the waterfall at Panau Falls Casino. It applies a moving force against you and whatever vehicle you're in, and it totalled my agency boat before reaching the bottom by bashing it against the cliff face. Might give you a little more insight into how it works?